Superbarrio Gómez ran the most underreported campaign of the 1996 U.S. presidential election. The masked Mexican wrestler turned social activist showed up in New Hampshire during the primary season and declared himself a “candidate” even though his foreign citizenship rendered him ineligible. Decisions affecting the lives of Mexicans are made in the White House, he reasoned, so Mexicans should have a say in choosing its occupant. It’s a sentiment that’s widely shared: two-thirds of the 26,000 respondents from 32 countries in a recent poll believe that the White House has an important impact on their lives, and for that reason, almost half believe they should have a vote in the U.S. presidential election. (If they did, President Barack Obama would be a shoo-in, according to almost every poll.)

The level of interest in this U.S. election, however, is considerably lower than that of 2008. One reason for this may be that any global citizen tuning in to the campaign’s foreign policy debate would have struggled to find substantial differences between what Governor Mitt Romney advocated and what the White House is doing. There is also a growing sense of the relative decline of U.S. global power. The U.S. remains the world’s most militarily powerful country and its largest economy, but its ability to shape economic and geopolitical events in distant climes has steadily declined over the past decade — whether it’s Afghanistan or Iraq, the rapidly changing Arab world or Europe’s debt crisis, Washington struggles to impose its will.

That said, the Oval Office remains the world’s strongest single center of power, and the outcome of Tuesday’s vote will be closely watched. Here are five places where the stakes are particularly high:

1. Syria: Breaking a Stalemate?

Syria’s civil war has killed upwards of 20,000 people, but the country remains locked in an effective stalemate: the regime of President Bashar Assad is unable to destroy the rebels, and the rebels are unable to destroy the regime. Given the sectarian stakes and the danger of igniting a wider regional conflict, as well as a fear of the growing influence of extremist elements among the rebels, the U.S. has held back from direct intervention, and even from enabling the rebels to receive heavier weaponry that could neutralize some of the regime’s military advantages. Both Obama and Romney say they oppose any direct deployment of U.S. military force in Syria, even if limited to enforcing a “no-fly zone” or protecting a rebel-controlled enclave — much to the frustration of allies such as Turkey and France. The Obama Administration is currently focused on forging a single opposition leadership structure based on a moderate consensus, in order to enable greater foreign support to the rebel cause. Many opposition activists and rebel fighters have expressed dismay and anger at what they see as the relative passivity of the Obama Administration in the face of the increasingly bloody impasse. As a result, the fact that Romney has publicly declared a greater willingness to consider arming the rebels — together with the fact that some of his advisers have expressed more hawkish policies toward Assad and his regime’s key backers, Iran and Russia — may give many in the rebel camp reason to hope the Republican candidate prevails on Tuesday.

President Obama won the votes of 78% of Jewish Americans in the 2008 election, and despite the GOP’s best efforts to erode that advantage by painting the incumbent as somehow hostile to the state of Israel — a charge vehemently rejected by the Democrats as well as by a number of top Israeli officials — polls suggest Obama will again secure close to 70% of the Jewish vote, compared with around 25% for Romney. If anything, those numbers indicate that Israel is not the primary issue on which most Jewish Americans vote, since polling suggests that a clear majority of Israelis (around 52%) favor a Romney victory, compared with just 25% for Obama.

Israel, nonetheless, appears to be among the very few countries in the world whose citizens would prefer a Romney victory, perhaps as a result of tensions between Obama and Prime Minister Benjamin Netanyahu over the President’s 2009 efforts to restart the peace process with the Palestinians. Comments made by Romney in a secretly recorded address to donors in Tampa earlier this year (he expressed doubt that any solution to the Israeli-Palestinian conflict is imminent, blamed the impasse on the Palestinians and vowed to kick the can down the road) certainly accord with mainstream Israeli thinking. But Romney later expressed his desire to see a two-state solution negotiated by the two sides. It remains unclear what either candidate would do to restart the moribund peace process if elected.

On the Israeli government’s primary concern, Iran, Romney has used more forceful language, but his effective policy commitments — sanctions backed by the threat of military action — appear broadly similar to Obama’s. Still, Romney’s drawing the line at Iran having the capability to build nuclear weapons rather than at actually starting the process, and a statement by one of his top aides that the GOP candidate would respect an Israeli decision to use force against Iran are deemed more pleasing to Israeli hawks. Then again, it’s generally understood in the Israeli strategic establishment that Israel’s optimal scenario is not a U.S. green light for an Israeli attack on Iran but for the U.S. to do the job. And on that front, Romney has also made clear that he does “not believe that in the final analysis we will have to use military action” against Iran.

It’s widely assumed, however, that efforts to find a diplomatic solution to the increasingly dangerous impasse will resume after the U.S. election, and Israel’s hawkish government will likely see a Romney Administration as more open to Israeli persuasion to adopt a hard line in any talks with Iran.

Tony Karon is a senior editor at TIME, where he has covered international conflicts in the Middle East, Asia, and the Balkans since 1997. A native of South Africa, he now resides with his family in Brooklyn, New York.